Inequalities Concerning the Lp Norm of a Polynomial and Its Derivative

Let Pn(z) = an ∏nv = 1 (z − zv), an ≠ 0, be a polynomial of degree n. It has been proved that if |zv| ≥ Kv ≥ 1, 1 ≤ v ≤ n, then for p ≥ 1, [formula] where Fp = {2π/∫2π0 |t0 + eiΘ)pdΘ}1/p and t0 = {1 + n/∑nv=1 (1/(Kv − 1))}. This result generalizes the well known Lp inequality due to De Bruijn for polynomials not vanishing in |z| < 1. On making p → ∞, it gives the L∞ inequality due to Govil and Labelle which as a special case includes the Erdős conjecture proved by Lax.